where \(p(x|\lambda )\)  is a probability distribution (which we hereafter refer to as a state) that depends on some parameterization, \(\vec{\lambda }\) , of the state space. \(F_{\mu \nu }\)  quantifies how distinguishable the distribution \(p(x|\vec{\lambda })\)  is from the distribution, \(p(x|\lambda + \lambda )\)  [1]}.
The quantum generalization holds for both pure [2]} and mixed [3]} states, and is termed quantum Fisher information (QFI) with the density matrix \(\hat{\rho }\)  taking the place of \(p(x|\lambda )\) , and the quantum expectation value replacing the integral.
The associated geometrical structure is termed quantum information geometry or simply
quantum geometry (see  [4]} for a complete introduction to the subject). In this work we are specifically interested in the case of pure states of spin-S with unitary parameterizations, for which the QFI matrix is proportional to the connected covariance [5]},
\(Q_{\mu \nu } \lbrace \psi \rbrace  = \frac{A}{S^2} \text{Cov}_\psi (\Lambda _\mu ,\Lambda _\nu )\) 
